RESONANT NORMAL FORMS, INTERPOLATING HAMILTONIANS AND STABILITY ANALYSIS OF AREA PRESERVING-MAPS

The geometrical and dynamical properties of area preserving maps in th e neighborhood of an elliptic fixed point are analyzed in the framewor k of resonant normal forms. The interpolating flow is not obtained fro m a map tangent to the identity, but from the normal form of the given map and a time independent interpolating Hamiltonian H is introduced. On this Hamiltonian the local stability properties of the fixed point and the geometric structure of the orbits are transparent. Numerical agreement between the level lines of H and the orbits of the map sugge sts that the perturbative expansion of H is asymptotic. This is confir med by a rigorous error analysis, based on majorant series: the error for the normal form expansion grows as n! while the truncation error f or H also has a factorial growth and in a disc of radius r can be made exponentially small with 1/r. The boundary of the global stability do main is considered; for the quadratic map the identification with the inner envelope of the homoclinic tangle of the hyperbolic fixed point is strongly suggested by numerical evidence.